cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A057324 First member of a prime triple in a p^2 + p - 1 progression.

Original entry on oeis.org

2, 3, 11, 13, 53, 131, 233, 241, 281, 569, 659, 691, 761, 881, 1693, 2063, 2411, 2521, 2551, 2663, 2729, 2741, 2861, 3089, 4021, 4159, 4201, 4243, 4423, 4793, 6091, 7103, 7229, 7369, 7753, 7829, 8053, 8641, 8669, 9041, 9059, 9539, 9649, 9769, 10513
Offset: 1

Views

Author

Patrick De Geest, Aug 15 2000

Keywords

Comments

There exist no such triples of the form p^2 + p + 1 because each third member is always divisible by 3.
Subsequence of A053184. - Pierre CAMI, Sep 13 2013

Examples

			2 -> 2^2+2-1 = 5 -> 5^2+5-1 = 29 hence the prime triple (2,5,29).

Links

Crossrefs

Cf. A053184, A053185, A057325.

Programs

  • Mathematica
    fmpQ[n_]:=AllTrue[NestList[#^2+#-1&,n,2],PrimeQ]; Select[Prime[Range[ 1300]],fmpQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jul 08 2019 *)
  • PARI
    is(n)=for(k=1,3,if(!isprime(n),return(0));n=n^2+n-1);1 \\ Charles R Greathouse IV, Sep 13 2013

A155008 Primes p such that (p-a)*(p+a)-+a*p are primes,a=4.

Original entry on oeis.org

3, 5, 7, 11, 19, 29, 31, 59, 101, 139, 239, 271, 829, 1031, 1201, 1439, 1511, 1531, 2251, 2609, 3929, 4349, 4969, 5449, 5639, 5711, 5801, 5881, 5981, 6521, 6569, 6701, 6949, 6959, 8221, 8831, 9001, 9181, 9209, 9419, 9511, 9929, 10139, 10711, 11839, 11981
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jan 18 2009

Keywords

Comments

3*11-28=5, 3*11+28=61, ...

Crossrefs

Cf. A125272, A053184, A038872, A141158, A038615, A098058, A038936, A089270, A140559, A154939, A155006, A155007

Programs

  • Mathematica
    lst={};Do[p=Prime[n];If[PrimeQ[(p-4)*(p+4)-4*p]&&PrimeQ[(p-4)*(p+4)+4*p],AppendTo[lst,p]],{n,7!}];lst
Select[Prime[Range[1500]],AllTrue[(#-4)(#+4)+{4#,-4#},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 27 2020 *)

A155009 Primes p such that (p-a)*(p+a)-+a*p are primes,a=5.

Original entry on oeis.org

2, 7, 11, 17, 19, 23, 41, 43, 61, 67, 107, 109, 131, 137, 179, 197, 263, 269, 331, 353, 397, 641, 677, 743, 859, 941, 1163, 1171, 1213, 1303, 1319, 1433, 1447, 1453, 1543, 1601, 1783, 2221, 2351, 2371, 2417, 2503, 2657, 2689, 2791, 2797, 2909, 3037, 3301
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jan 18 2009

Keywords

Comments

1*12-35=-23, 1*12+35=47; 6*16-55=96-55=41, 6*16-55=96+55=151, ...

Links

Crossrefs

Cf. A125272, A053184, A038872, A141158, A038615, A098058, A038936, A089270, A140559, A154939, A155006, A155007, A155008

Programs

  • Mathematica
    lst={};Do[p=Prime[n];If[PrimeQ[(p-5)*(p+5)-5*p]&&PrimeQ[(p-5)*(p+5)+5*p],AppendTo[lst,p]],{n,7!}];lst
Select[Prime[Range[500]],AllTrue[(#-5)(#+5)+{5#,-5#},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 01 2016 *)

A226856 Numbers A226770((prime(n))^2-1).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 3, 1, 3, 5, 1, 46, 1, 5, 66, 1, 1, 3, 3, 3, 10, 3, 1, 1, 172, 1, 1, 398, 5, 132, 255, 1, 336, 41, 1, 3, 615, 1, 462, 918, 283, 1, 1, 1, 453, 1, 11, 713, 1503, 311, 1, 112, 1, 308, 2272, 1, 788, 481, 2301, 1, 2686, 1358, 3, 3, 2651, 3346, 1
Offset: 1

Views

Author

Vladimir Shevelev, Jun 19 2013

Keywords

Comments

a(n) = 1 iff (prime(n))^2 + prime(n) - 1 is prime (A053184).

Links

Crossrefs

Cf. A226770, A053184.

Programs

  • Mathematica
    Table[(div=Most[Divisors[n+1]]; Length[FixedPoint[Union[Flatten[AppendTo[div, Map[Most[Rest[Divisors[n+#]]]&, #]]]]&, div]]-1), {n, Prime[Range[40]]^2-1}] (* Peter J. C. Moses, Jun 20 2013 *)

Extensions

More terms from Peter J. C. Moses, Jun 20 2013

A226859 Number of prime sums in the process described in A226770.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 3, 1, 3, 2, 3, 1, 4, 1, 4, 2, 5, 1, 5, 1, 6, 3, 7, 1, 6, 1, 7, 4, 7, 3, 8, 1, 9, 4, 9, 1, 9, 1, 9, 4, 10, 1, 9, 2, 10, 2, 11, 1, 11, 2, 13, 5, 14, 1, 13, 1, 12, 5, 12, 5, 13, 1, 13, 6, 14, 1, 14, 1, 13, 6, 14, 7, 15, 1, 15, 3, 15
Offset: 1

Views

Author

Vladimir Shevelev, Jun 20 2013

Keywords

Examples

			Let n=76. We have 77; d=7,11; 76+7=83 (prime), 76+11=87; d=3,29; 76+3=79(prime), 76+29=105; d=5,15,21,35; 76+5=81, 76+15=91, 76+21=97(prime), 76+35=111; d=9,27,13,37, 76+9=85,76+27=103(prime),76+13=89(prime), 76+37=113(prime), d=17, 76+17=93; d=31, 76+31=107(prime). Thus the set of prime sums is {83,79,97,103,89,113,107} and therefore a(76)=7.

Links

Crossrefs

Cf. A226770, A226856, A053184.

Programs

Formula

a(n) = 1 iff either n = 5 or n + 1 = p or n + 1 = q^2, where p,q and q^2+q-1 are primes.

Extensions

More terms from Peter J. C. Moses, Jun 20 2013

A243471 Primes p such that p^6 - p^5 + 1 is prime.

Original entry on oeis.org

3, 31, 73, 181, 367, 373, 523, 631, 733, 1021, 1039, 1171, 1489, 1723, 1777, 2203, 2557, 2683, 3121, 3187, 3319, 4441, 4591, 4621, 4801, 4957, 5113, 5167, 5323, 5431, 5659, 5839, 5851, 5857, 6883, 7057, 7129, 7297, 7309, 7477, 7993, 8017, 8209, 8221, 8689, 8821
Offset: 1

Views

Author

K. D. Bajpai, Jun 05 2014

Keywords

Examples

			31 appears in the sequence because it is prime and 31^6 - 31^5  + 1 = 858874531 is also prime.
73 appears in the sequence because it is prime and 73^6 - 73^5  + 1 = 149261154697 is also prime.

Links

Crossrefs

Cf. A000040, A091567, A053182, A053183, A053184, A238136.

Programs

  • Maple
    A243471 := proc() local a, b; a:=ithprime(n); b:= a^6-a^5+1; if isprime (b) then RETURN (a); fi; end: seq(A243471 (), n=1..2000);
  • Mathematica
    c=0; Do[k=Prime[n]; If[PrimeQ[k^6-k^5+1], c++; Print[c," ",k]], {n,1,200000}];

A243472 Primes p such that p^6 - p^5 - 1 is prime.

Original entry on oeis.org

2, 31, 101, 151, 181, 199, 229, 277, 307, 317, 379, 439, 479, 491, 647, 691, 797, 911, 997, 1039, 1051, 1181, 1291, 1367, 1381, 1471, 1511, 1549, 1657, 1709, 1847, 1867, 1987, 2081, 2099, 2111, 2207, 2467, 2621, 2707, 3041, 3221, 3259, 3541, 3571, 3581, 3769
Offset: 1

Views

Author

K. D. Bajpai, Jun 05 2014

Keywords

Examples

			31 appears in the sequence because it is prime and 31^6 - 31^5 - 1 = 858874529 is also prime.
101 appears in the sequence because it is prime and 101^6 - 101^5  - 1 = 1051010050099 is also prime.

Links

Crossrefs

Cf. A000040, A091567, A053182, A053183, A053184, A238136.

Programs

  • Maple
    A243472 := proc() local a, b; a:=ithprime(n); b:= a^6-a^5-1; if isprime (b) then RETURN (a); fi; end: seq(A243472 (), n=1..2000);
  • Mathematica
    c = 0;  Do[k=Prime[n]; If[PrimeQ[k^6-k^5-1], c++; Print[c," ",k]], {n,1,200000}];
Select[Prime[Range[600]],PrimeQ[#^6-#^5-1]&] (* Harvey P. Dale, Jan 21 2015 *)
  • PARI
    s=[]; forprime(p=2, 4000, if(isprime(p^6-p^5-1), s=concat(s, p))); s \\ Colin Barker, Jun 06 2014

A136243 Numbers k in A008864 such that k^2 - k - 1 is prime.

Original entry on oeis.org

3, 4, 6, 12, 14, 20, 32, 42, 54, 60, 84, 90, 102, 104, 132, 150, 164, 182, 192, 194, 200, 234, 242, 264, 282, 332, 350, 374, 402, 420, 432, 434, 450, 462, 464, 500, 542, 570, 572, 660, 674, 684, 692, 710, 740, 744, 762, 770, 810, 864, 882, 942, 1014, 1040
Offset: 1

Views

Author

Lekraj Beedassy, Dec 24 2007

Keywords

Comments

See A053185 for the primes associated with a(n).

Crossrefs

Cf. A008864, A053184, A136242.

Programs

  • PARI
    isok(k) = isprime(k-1) && isprime(k^2-k-1); \\ Michel Marcus, Dec 19 2022

Formula

a(n) = A053184(n) + 1.

Extensions

a(20)=194 inserted by Georg Fischer, Dec 18 2022

A154942 Primes p such that (p-1)*p*(p+1)-p-2 and (p-1)*p*(p+1)+p+2 are primes.

Original entry on oeis.org

3, 5, 29, 71, 113, 263, 1103, 2339, 3203, 3413, 3593, 3659, 3719, 4421, 5939, 6269, 7841, 9011, 9029, 13121, 13841, 14423, 15671, 17033, 19073, 22079, 22811, 26783, 27851, 28949, 29303, 30839, 31973, 32063, 32141, 34301, 38543, 38873, 39119
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jan 17 2009

Keywords

Comments

2*3*4=24-3-2=19, 2*3*4=24+3+2=29, ...

Crossrefs

Cf. A053184, A038872, A141158, A038615, A098058, A038936, A089270, A140559, A154939

Programs

  • Mathematica
    lst={};Do[p=Prime[n];If[PrimeQ[(p-1)*p*(p+1)-p-2]&&PrimeQ[(p-1)*p*(p+1)+p+2],AppendTo[lst,p]],{n,8!}];lst
prQ[n_]:=Module[{x=n^3-n,y=n+2},And@@PrimeQ[{x+y,x-y}]]; Select[Prime[ Range[4200]],prQ] (* Harvey P. Dale, Jun 21 2012 *)

A243544 Primes p such that p^2 - p + 1 is semiprime.

Original entry on oeis.org

5, 11, 29, 37, 41, 43, 53, 61, 71, 73, 83, 97, 109, 113, 127, 137, 149, 157, 167, 181, 191, 211, 223, 229, 241, 271, 277, 281, 307, 317, 331, 359, 389, 421, 433, 443, 461, 463, 487, 499, 547, 557, 571, 587, 601, 617, 631, 659, 661, 683, 691, 701, 709, 733, 757
Offset: 1

Views

Author

K. D. Bajpai, Jun 06 2014

Keywords

Comments

Intersection of A000040 and A180748.

Examples

			11 is in the sequence because 11 is prime and 11^2 - 11 + 1 = 111 = 3 * 37 is semiprime.
29 is in the sequence because 29 is prime and 29^2 - 29 + 1 = 813 = 3 * 271 is semiprime.
17 is not in the sequence though 17 is prime, because 17^2 - 17 + 1 = 273 = 3 * 7 * 13, has more than two prime factors.

Links

Crossrefs

Cf. A000040, A001358, A053182, A053184, A065508, A091567, A180748.

Programs

  • Maple
    with(numtheory): A243544 := proc() local a; a:=ithprime(n);  if bigomega(a^2-a+1)=2 then RETURN (a); fi; end: seq(A243544 (), n=1..200);
  • Mathematica
    c = 0; Do[k = Prime[n]; If[PrimeOmega[k^2 - k + 1] == 2, c++; Print[c, " ", k]], {n, 1, 30000}];
Select[Prime[Range[150]],PrimeOmega[#^2-#+1]==2&] (* Harvey P. Dale, Oct 22 2024 *)
  • PARI
    s=[]; forprime(p=2, 800, if(bigomega(p^2-p+1)==2, s=concat(s, p))); s \\ Colin Barker, Jun 06 2014
Previous Showing 11-20 of 25 results. Next

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